Magnetic Gradient and Curvature Based Ranging Method

ABSTRACT

Methods for determining a distance from a drilling well to a magnetized target well include acquiring magnetic field measurements from the drilling well. The acquired magnetic field measurements are made at a plurality of spaced apart locations in the drilling well. The acquired magnetic field measurements are processed to obtain a ratio including at least one of the following: (i) a ratio of a magnetic field intensity to a first spatial derivative of a magnetic field, (ii) a ratio of a magnetic field intensity to a second spatial derivative of a magnetic field, and (iii) a ratio of a first spatial derivative of a magnetic field to a second spatial derivative of the magnetic field. The ratio (or ratios) is then processed to obtain the distance from the drilling well to the magnetized target well.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority to and the benefit of U.S. ProvisionalPatent Application No. 61/894,460, filed 24 Oct. 2013, which isincorporated by reference herein.

FIELD OF THE INVENTION

Disclosed embodiments relate generally to drilling and surveyingsubterranean boreholes such as for use in oil and natural gasexploration and more particularly to methods for determining a distancebetween a twin well and a magnetized target well using first spatialderivatives and second spatial derivatives of a measured magnetic field.

BACKGROUND INFORMATION

Magnetic ranging measurements may be used to obtain a distance and adirection to an adjacent well. For example, commonly assigned U.S. Pat.No. 7,656,161 discloses a technique in which a predetermined magneticpattern is deliberately imparted to a plurality of casing tubulars.These tubulars, thus magnetized, are coupled together and lowered intothe adjacent well (the target well) to form a magnetized section ofcasing string typically including a plurality of longitudinally spacedpairs of opposing magnetic poles. Measurements of the magnetic field maythen be utilized to survey and guide drilling of a drilling well (e.g. atwin well) relative to the target well. The distance between the twinand target wells may be determined from various magnetic fieldmeasurements made in the twin well (as further disclosed in commonlyassigned U.S. Pat. No. 7,617,049). These well twinning techniques may beadvantageously utilized, for example, in steam assisted gravity drainage(SAGD) applications in which horizontal twin wells are drilled torecover heavy oil from tar sands.

While the above described methodology has been successfully utilized inwell twinning applications, there is room for yet further improvement.For example, it can be difficult to accurately remove the earth'smagnetic field from the measured magnetic field since the attitude ofthe drilling well is not generally known with precision. Moreover, sincethe distance between the two wells is obtained from the measuredmagnetic field strength (intensity), any changes in the strength of thecasing magnetization may cause a corresponding error in the obtaineddistance (e.g., a decay in the casing magnetization may cause thedistance to be underestimated). Therefore there is a need for improvedranging methodologies.

SUMMARY

Methods for determining a distance from a drilling well to a magnetizedtarget well are disclosed. The methods include acquiring magnetic fieldmeasurements from the drilling well. A drill string is deployed in thedrilling well and includes at least one magnetic field sensor in sensoryrange of magnetic flux emanating from the magnetized target well. Theacquired magnetic field measurements are made at a plurality of spacedapart locations, e.g., at a plurality of spaced apart axial and/orradial locations in the drilling well. The acquired magnetic fieldmeasurements are processed to obtain a ratio including at least one ofthe following: (i) a ratio of a magnetic field intensity to a firstspatial derivative of a magnetic field, (ii) a ratio of a magnetic fieldintensity to a second spatial derivative of a magnetic field, and (iii)a ratio of a first spatial derivative of a magnetic field to a secondspatial derivative of the magnetic field. The ratio (or ratios) are thenprocessed to obtain the distance from the drilling well to themagnetized target well.

The disclosed embodiments may provide various technical advantages. Forexample, the disclosed methods may improve the accuracy of the distancesdetermined via magnetic ranging by reducing the dependence of themagnetic ranging measurements on the strength of the targetmagnetization. Moreover, certain of the disclosed embodiments mayobviate the need to remove the earth's magnetic field from the measuredmagnetic field.

This summary is provided to introduce a selection of concepts that arefurther described below in the detailed description. This summary is notintended to identify key or essential features of the claimed subjectmatter, nor is it intended to be used as an aid in limiting the scope ofthe claimed subject matter.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the disclosed subject matter, andadvantages thereof, reference is now made to the following descriptionstaken in conjunction with the accompanying drawings, in which:

FIG. 1 depicts a prior art arrangement for a SAGD well twinningoperation.

FIG. 2 depicts a prior art magnetization of a wellbore tubular.

FIG. 3 depicts a flow chart of one example of a disclosed methodembodiment for determining a distance between a drilling well and amagnetized target well.

FIG. 4 depicts a plot of the magnetic field about a magnetized casingstring.

FIGS. 5A and 5B depict plots of the axial and radial components (B_(z)and B_(r)) of the magnetic field as a function of normalized axialposition along the target well at various distances from the targetwell.

FIGS. 6A, 6B, and 6C depict plots of the three independent first spatialderivatives of the magnetic field as a function of normalized axialposition along the target well at various distances from the targetwell.

FIGS. 7A, 7B, 7C, and 7D depict plots of the four independent secondspatial derivatives of the magnetic field as a function of normalizedaxial position along the target well at various distances from thetarget well.

FIGS. 8A and 8B depict plots of various ratios of a magnetic fieldintensity to a first spatial derivative of the magnetic field as afunction of the actual distance to the magnetized target.

FIGS. 9A and 9B depict plots of various ratios of a magnetic fieldintensity to a second spatial derivative of the magnetic field as afunction of the actual distance to the magnetized target.

FIGS. 10A, 10B, 10C, and 10D depict plots of various ratios of a firstspatial derivative of a magnetic field to a second spatial derivative ofthe magnetic field as a function of the actual distance to themagnetized target.

DETAILED DESCRIPTION

FIG. 1 schematically depicts one example of a well twinning applicationsuch as a SAGD twinning operation. Common SAGD twinning operationsrequire a horizontal twin well 20 to be drilled a substantially fixeddistance substantially directly above a horizontal portion of the targetwell 30 (e.g., not deviating more than about 1-2 meters up or down or tothe left or right of the lower well). In the exemplary embodiment shown,the lower (target) well 30 is drilled first, for example, usingconventional directional drilling and MWD techniques. However, thedisclosed embodiments are not limited in regard to which of the wells isdrilled first. The target wellbore 30 is then cased using a plurality ofpremagnetized tubulars (such as those shown on FIG. 2 described below)to form a magnetized casing string 35. In the embodiment shown, drillstring 24 includes at least one tri-axial magnetic field measurementsensor 28 deployed in close proximity to the drill bit 22. Sensor 28 isused to passively measure the magnetic field about target well 30 as thetwin well is drilled. Such passive magnetic field measurements are thenutilized to guide continued drilling of the twin well 20 along apredetermined path relative to the target well 30 (e.g., as described inU.S. Pat. Nos. 7,617,049, 7,656,161, and 8,026,722, each of which isfully incorporated by reference herein).

With reference now to FIG. 2, one example tubular 60 magnetized asdescribed in the '722 patent is shown. The depicted tubular 60embodiment includes a plurality of discrete magnetized zones 62(typically three or more). Each magnetized zone 62 may be thought of asa discrete cylindrical magnet having a north N pole on one longitudinalend thereof and a south S pole on an opposing longitudinal end thereofsuch that a longitudinal magnetic flux 68 is imparted to the tubular 60.Tubular 60 further includes a single pair of opposing north-north NNpoles 65 at the midpoint thereof. The purpose of the opposing magneticpoles 65 is to focus magnetic flux outward from tubular 60 as shown at70 (or inward for opposing south-south poles as shown at 72). Thetubulars may be magnetized, for example, using the apparatus disclosedin U.S. Pat. No. 7,538,650, which is fully incorporated by referenceherein.

With continued reference to FIG. 1, the casing string 35 is formed byjoining (threadably connecting) premagnetized tubulars in the targetwell 30. In one embodiment, the resultant string 35 has a single pair ofopposing magnetic poles in the central region (the middle third) of eachtubular. Thus the pairs of opposing magnetic poles (NN or SS) are spacedat intervals about equal to the length of tubulars, while the period ofthe magnetic field pattern (e.g., the distance from one a NN pair ofopposing magnetic poles to the next NN pair) is about twice the lengthof a tubular.

As described above, drill string 20 may include a triaxial magneticfield sensor 28. The depicted embodiment of the sensor 28 includes threemutually orthogonal magnetic field sensors, one of which is orientedsubstantially parallel with the borehole axis (M_(Z)). Sensor 28 maythus be considered as determining a plane (defined by M_(X) and M_(Y))orthogonal to the borehole axis and a pole (M_(Z)) parallel to theborehole axis of the twin well, where M_(X), M_(Y), and M_(Z) representmeasured magnetic field vectors in the x, y, and z directions.

The magnetic field about the magnetized casing string may be measuredand represented, for example, as a vector whose orientation depends onthe location of the measurement point within the magnetic field. Inorder to determine the magnetic field vector due to the target well(e.g., target well 30) at any point downhole, the magnetic field of theearth may be subtracted from the measured magnetic field vector usingmeans known to those of ordinary skill in the art. The magnetic field ofthe earth (including both magnitude and direction components) may beknown, for example, from previous geological survey data or ageomagnetic model. It will be understood that in certain embodimentssuch subtraction of the magnetic field of the earth is not required.

It will be appreciated that the disclosed embodiments are not limited tothe depictions of FIGS. 1 and 2. For example, the disclosure is notlimited to SAGD applications. Rather, exemplary methods in accordancewith this disclosure may be utilized to drill twin wells havingsubstantially any relative orientation for substantially anyapplication. Moreover, the disclosure is not limited to any particularmagnetization pattern or spacing of pairs of opposing magnetic poles onthe target well.

FIG. 3 depicts a flow chart of one example of a disclosed methodembodiment 100 for determining a distance between a drilling well and amagnetized target well (e.g., as depicted on FIG. 1). The methodincludes acquiring a plurality of axially and or radially spacedmagnetic field measurements at 110. The magnetic field measurements maythen be processed at 120 to compute first spatial derivatives and secondspatial derivatives of the magnetic field. The first spatial derivativesand second spatial derivatives may be further processed at 130 tocompute one or more of the following ratios: (i) a ratio of the magneticfield intensity to a first spatial derivative of the magnetic field,(ii) a ratio of the magnetic field intensity to a second spatialderivative of the magnetic field, and/or (iii) a ratio of a firstspatial derivative of the magnetic field to a second spatial derivativeof the magnetic field. The computed ratio or ratios may then be furtherprocessed to obtain the distance between the drilling well and themagnetized target well at 140.

The plurality of axial and/or radially spaced magnetic fieldmeasurements may be acquired at 110 using magnetic field sensorsdeployed in a drill string in the drilling well (e.g., sensor 28deployed in drill string 24 in drilling well 20 in FIG. 1). In certainembodiments, the spaced magnetic field measurements may be made using asingle triaxial magnetic field sensor. For example, axially spacedmeasurements may be obtained via moving the drill string axially in thewellbore (in the uphole or downhole direction) between measurements.Radially spaced measurements may be obtained by rotating an off-centered(eccentered) sensor to various toolface angles between measurements. Inother embodiments, the drill string may include a plurality of axiallyand/or radially spaced magnetic field sensors. For example, two, three,or more axially spaced measurements may be acquired via correspondingmagnetic field sensors deployed in the drill string (e.g., at half meterintervals along the length of the string). Radially spaced measurementsmay be acquired via corresponding magnetic field sensors deployed aboutthe circumference of the drill string (e.g., first and seconddiametrically opposed sensors or three or more sensors deployed atsuitable angular intervals about the circumference). Radially spacedmeasurements may also be acquired using corresponding sensors havingdifferent degrees of eccentricity (e.g., a central sensor and one ormore eccentered sensors). The magnetic field sensors may also be offsetboth axially and radially (e.g., first and second axially spaced sensorshaving one or more eccentered sensors located axially between them). Thedisclosed method embodiments are not limited to any particular magneticfield sensor configuration and/or spacing.

The magnetic field measurements may be resolved into three orthogonalcomponents which can in turn be defined, for example, as highside,lateral, and along-hole or axial directions (or x, y, and z directionsas described above). The highside and lateral components may also beresolved into polar coordinates, designated, for example, by a radialintensity and a toolface-to-target direction. Four magnetic fieldgradients (first spatial derivatives of the magnetic field) may bedefined based on the axial and radial components. However, since themagnetic field is magnetostatic and current-free, its curl is zero andonly three of these gradients are independent as indicated below:

$\begin{matrix}{{\frac{\partial B_{r}}{\partial r};}{\frac{\partial B_{z}}{\partial z};}{and}{\frac{\partial B_{r}}{\partial z} = \frac{\partial B_{z}}{\partial r}}} & (1)\end{matrix}$

where B_(z) and B_(r) represent the intensity of the measured magneticfield in the axial (z) and radial (r) directions. Four independentsecond spatial derivatives of the magnetic field may also be obtainedbased on the axial and radial components of the magnetic field. They areas follows:

$\begin{matrix}{{\frac{\partial^{2}B_{r}}{\partial r^{2}};}{\frac{\partial^{2}B_{z}}{\partial z};}{{\frac{\partial^{2}B_{r}}{\partial z^{2}} = \frac{\partial^{2}B_{z}}{{\partial r} \cdot {\partial z}}};}{and}{\frac{\partial^{2}B_{z}}{\partial r^{2}} = \frac{\partial^{2}B_{r}}{{\partial r} \cdot {\partial z}}}} & (2)\end{matrix}$

It will be understood that at least two spaced apart magnetic fieldmeasurements are generally required to obtain a first spatial derivativeof the magnetic field (a gradient of the magnetic field) and that atleast three spaced apart magnetic field measurements are generallyrequired to obtain a second spatial derivative of the magnetic field (acurvature of the magnetic field).

The magnetic field gradients may be computed at 120, for example, fromfirst and second spaced apart magnetic field measurements. For example,the gradient of the axial component of the magnetic field in the axialdirection (∂B_(z)/∂z) may be obtained as follows:

$\begin{matrix}{\frac{\partial B_{z}}{\partial z} = \frac{\Delta \; B_{z}}{\Delta \; z}} & (3)\end{matrix}$

where ΔB_(z) represents the difference in the axial component of themagnetic field between the first and second measurement positions (i.e.,ΔB_(z)=B_(z2)−B_(z1)) and Δz represents the axial measurement spacing(the distance between the first and second measurement positions, i.e.,Δz=z₂−z₁). Gradients of the radial component of the magnetic fieldand/or in the radial direction may be similarly computed.

The second spatial derivatives may be computed at 120, for example, fromfirst, second, and third spaced apart magnetic field measurements. Forexample, the curvature of the axial component of the magnetic field inthe axial direction (∂² B_(z)/∂z²) may be obtained as follows:

$\begin{matrix}{\frac{\partial^{2}B_{z}}{\partial z^{2}} = {\frac{\left( {{\frac{\Delta \; B_{z}}{\Delta \; z}(2)} - {\frac{\Delta \; B_{z}}{\Delta \; z}(1)}} \right)}{\Delta \; z} = \frac{B_{z\; 3} - {2B_{z\; 2}} + B_{z\; 1}}{\left( {\Delta \; z} \right)^{2}}}} & (4)\end{matrix}$

where

$\frac{\Delta \; B_{z}}{\Delta \; z}(1)$

represents the magnetic field gradient between the first and secondaxial positions,

$\frac{\Delta \; B_{z}}{\Delta \; z}(2)$

represents the magnetic field gradient between the second and thirdaxial positions, and Δz represents the axial measurement spacing. Thesecond spatial derivatives may also be obtained, for example, by fittingthree or more spaced measurements to a function such as a polynomial andthen differentiating the function. Second spatial derivatives of theradial component of the magnetic field and/or in the radial directionmay be similarly computed.

Owing to the dimensional constraints on downhole tools, the radialmeasurement spacing tends to be limited to about 0.1 meters or less. Thespacing in the axial direction is not physically constrained in the sameway; however, it may be advantageous for the axial measurement spacingto be less than about a few meters in order to maintain good resolutionand to avoid complications caused by tool curvature. The short radialmeasurement spacing tends to increases sensitivity to noise such that incertain operations it may be advantageous to use the axially distributedmeasurements ∂B_(r)/∂z, ∂B_(z)/∂z, ∂²B_(r)/∂z², and ∂²B_(z)/∂z² whenpossible.

Variations in the first spatial derivatives and the second spatialderivatives of the magnetic field with position relative to a magnetizedtarget well may be evaluated using a magnetic model. For example, amagnetized casing string having a repeating magnetic pattern along theaxis of the string (e.g., as described above with respect to FIGS. 1 and2) may be modelled as a repeating series of point sources (monopoles)and/or line sources distributed along the centerline of the string. Fora monopole model, the field at any point (r, z) from a point sourcelocated at (0, zp) may expressed as follows:

$\begin{matrix}{B_{z} = {\frac{P}{4\pi} \cdot \frac{\left( {z - {zp}} \right)}{\left\lbrack {\left( {z - {zp}} \right)^{2} + r^{2}} \right\rbrack^{1.5}}}} & (5) \\{B_{r} = {\frac{P}{4\pi} \cdot \frac{r}{\left\lbrack {\left( {z - {zp}} \right)^{2} + r^{2}} \right\rbrack^{1.5}}}} & (6)\end{matrix}$

where P represents the strength of each of the magnetic poles and 0≦p<1and represents the axial location along the repeating magnetic pattern(where the positions p=0, 1, . . . are adjacent NN opposing magneticpoles). For a line source model, the field at any point (r, z) from aline source of length L centered at (0, zp) may expressed as follows:

$\begin{matrix}{B_{z} = {\frac{P}{4\; \pi \; L} \cdot \begin{bmatrix}{\frac{1}{\sqrt{\left( {z - {zp} - {L/2}} \right)^{2} + r^{2}}} -} \\\frac{1}{\sqrt{\left( {z - {zp} + {L/2}} \right)^{2} + r^{2}}}\end{bmatrix}}} & (7) \\{B_{r} = {\frac{P}{4\; \pi \; {Lr}} \cdot \begin{bmatrix}{\frac{z - {zp} + {L/2}}{\sqrt{\left( {z - {zp} + {L/2}} \right)^{2} + r^{2}}} -} \\\frac{z - {zp} - {L/2}}{\sqrt{\left( {z - {zp} - {L/2}} \right)^{2} + r^{2}}}\end{bmatrix}}} & (8)\end{matrix}$

FIG. 4 depicts a plot of the actual magnetic field about a magnetizedcasing string. The field is represented as a plot of the axial componentof the magnetic field versus the radial component of the magnetic field.The magnetic field is further plotted at various radial distances fromthe string. The casing string was magnetized with a repeating pattern ofopposing magnetic poles such that the pattern repeats with a period oftwice the length of the tubulars that make up the string (as describedabove). It may be noted that the casing magnetization in this example ismildly asymmetric with the left side of the plot being larger than theright side, possibly indicating that joints magnetized with one polarityretained slightly more magnetization than the others (the disclosedembodiments are of course not limited in this regard). This fact willaid in determining the sensitivity of a ranging technique to theabsolute magnetization of the target as ideally the calculated distanceshould be the same for both joints.

When ranging to a target well magnetized as described above, drillingmay be stopped and magnetic surveys taken at locations corresponding tomaximum radial flux from the target (i.e. adjacent the NN or SS opposingmagnetic poles located at the approximate midpoint of each tubular). Atthese locations the axial field from the target tends to be small (nearzero) while the radial field tends to be at a maximum. These locationscorrespond to the left and right sides of the plot depicted on FIG. 4.The gradients ∂B_(z)/∂z and ∂B_(r)/∂r are relatively large at theselocations while ∂B_(r)/∂z is small (near zero). Of the second spatialderivatives, ∂²B_(r)/∂r² and ∂²B_(r)/∂z² tend to be large, while∂²B_(z)/∂r² and ∂²B_(z)/∂z² are small (near zero). Since measurements ofsmall quantities tend to be susceptible to noise, it may be advantageousmake use of the larger values ∂B_(r)/∂r, ∂B_(z)/∂z, ∂²B_(r)/∂r², and∂²B_(r)/∂z², and particularly the long baseline measurements ∂B_(z)/∂zand ∂²B_(r)/∂z².

FIGS. 5A and 5B depict plots of the axial and radial components (B_(z)and B_(r)) of the magnetic field as a function of normalized axialposition along the target well at various distances from the targetwell. The joint ends are located at normalized axial positions of 1.0and 2.0 while the opposing magnetic poles are located at normalizedaxial positions of 0.5, 1.5, and 2.5 (with SS opposing magnetic polesbeing located at 0.5 and 2.5 and a NN opposing magnetic pole beinglocated at 1.5). Consistent with the plot depicted on FIG. 4, the radialcomponent has maxima at axial positions of 0.5, 1.5, and 2.5 (adjacentto the opposing magnetic poles).

FIGS. 6A, 6B, and 6C depict plots of the three independent magneticfield gradients (first spatial derivatives) as a function of normalizedaxial position along the target well at various distances from thetarget well. FIG. 6A depicts the gradient of the intensity of the radialmagnetic field component in the radial direction ∂B_(r)/∂r. FIG. 6Bdepicts the gradient of the intensity of the axial magnetic fieldcomponent in the axial direction ∂B_(z)/∂z. And FIG. 6C depicts thegradient of the intensity of the radial magnetic field component in theaxial direction ∂B_(r)/∂z (which is equal to the gradient of theintensity of the axial magnetic field component in the radial direction∂B_(z)/∂r). FIGS. 6A and 6B show that ∂B_(r)/∂r and ∂B_(z)/∂z havemaxima at axial positions of 0.5, 1.5, and 2.5 (adjacent the opposingmagnetic poles). FIG. 6C shows that ∂B_(r)/∂z is approximately zero atthe same axial positions.

FIGS. 7A, 7B, 7C, and 7D depict plots of the four independent secondspatial derivatives of the magnetic field as a function of normalizedaxial position along the target well at various distances from thetarget well. FIG. 7A depicts the second spatial derivative of the radialcomponent of the magnetic field in the radial direction ∂²B_(r)/∂r².FIG. 7B depicts the second spatial derivative of the radial component ofthe magnetic field in the axial direction ∂²B_(r)/∂z². FIG. 7C depictsthe second spatial derivative of the axial component of the magneticfield in the radial direction ∂²B_(z)/∂r². FIG. 7D depicts the secondspatial derivative of the axial component of the magnetic field in theaxial direction ∂²B_(z)/∂z². FIGS. 7A and 7B show that ∂²B_(r)/∂r² and∂²B_(r)/∂z² have maxima at axial positions of 0.5, 1.5, and 2.5(adjacent the opposing magnetic poles). FIGS. 7C and 7D show that∂²B_(z)/∂r² and ∂²B_(z)/∂z² are approximately zero at the same axialpositions.

When the magnetic field measurements are made at axial positionsadjacent (or nearly adjacent) to the opposing magnetic poles, themagnetic field intensity, the first spatial derivatives, and the secondspatial derivatives may be approximated, for example, from Equations 5and 6 above (the monopole approximation). Thus, for example, when z=zpthe magnetic field intensities may be expressed as follows:

$\begin{matrix}{B_{z} \approx 0} & (9) \\{B_{r} \approx \frac{P}{4\; \pi \; r^{2}}} & (10)\end{matrix}$

The first spatial derivatives may be also be expressed, for example asfollows:

$\begin{matrix}{\frac{\partial B_{z}}{\partial z} \approx \frac{P}{4\; \pi \; r^{3}}} & (11) \\{\frac{\partial B_{r}}{\partial r} \approx \frac{P}{4\; \pi \; r^{3}}} & (12) \\{\frac{\partial B_{r}}{\partial z} \approx \frac{\partial B_{z}}{\partial r} \approx 0} & (13)\end{matrix}$

The second spatial derivatives may also be expressed, for example, asfollows:

$\begin{matrix}{\frac{\partial^{2}B_{r}}{\partial r^{2}} \approx \frac{3P}{2\; \pi \; r^{4}}} & (14) \\{\frac{\partial^{2}B_{r}}{\partial z^{2}} \approx \frac{3P}{4\; \pi \; r^{4}}} & (15) \\{\frac{\partial^{2}B_{z}}{\partial r^{2}} \approx \frac{\partial^{2}B_{r}}{\partial z^{2}} \approx 0} & (16)\end{matrix}$

As described above, the intent of the magnetic ranging measurements isto determine the relative position of the drilling well with respect tothe magnetized target well, for example, via determining a distance anddirection from the drilling well to the target well. The toolfacedirection (the direction in the plane normal to the tool axis) towardsthe target may be obtained from a ratio of the two components measuredin that plane (e.g., a ratio of the x and y components of the measuredmagnetic field). The distance to the target may be found from a ratio ofa magnetic field intensity to a first spatial derivative of the magneticfield, a ratio of a magnetic field intensity to a second spatialderivative of the magnetic field, and/or a ratio of a first spatialderivative of the magnetic field to a second spatial derivative of themagnetic field. The use of one or more of the following ratios may beadvantageous in that the ratios are independent of the strength of themagnetic poles. The use of multiple ratios may further improve theaccuracy of the obtained distance by giving corresponding multipleindependent measurements.

When the magnetic field measurements are made at axial positionsadjacent (or nearly adjacent) to the opposing magnetic poles, the ratiosmay be approximated from certain of Equations 9 through 16 above. Thedistance to the target may be expressed in terms of example ratios of amagnetic field intensity to a first spatial derivative of the magneticfield, for example, as follows:

$\begin{matrix}{r \approx \frac{B_{r}}{\frac{\partial B_{z}}{\partial z}}} & (17) \\{r \approx {{- 2}\frac{B_{r}}{\frac{\partial B_{r}}{\partial r}}}} & (18)\end{matrix}$

The distance to the target may also be expressed in terms of exampleratios of a magnetic field intensity to a second spatial derivative ofthe magnetic field, for example, as follows:

$\begin{matrix}{r \approx \left\lbrack {6\frac{B_{r}}{\frac{\partial^{2}B_{r}}{\partial r^{2}}}} \right\rbrack^{\frac{1}{2}}} & (19) \\{r \approx \left\lbrack {{- 3}\frac{B_{r}}{\frac{\partial^{2}B_{r}}{\partial z^{2}}}} \right\rbrack^{\frac{1}{2}}} & (20)\end{matrix}$

The distance to the target may be further expressed in terms of exampleratios of a first spatial derivative of the magnetic field to a secondspatial derivative of the magnetic field, for example, as follows:

$\begin{matrix}{r \approx {6\frac{\frac{\partial B_{z}}{\partial z}}{\frac{\partial^{2}B_{r}}{\partial r^{2}}}}} & (21) \\{r \approx {{- 3}\frac{\frac{\partial B_{z}}{\partial z}}{\frac{\partial^{2}B_{r}}{\partial z^{2}}}}} & (22) \\{r \approx {{- 3}\frac{\frac{\partial B_{r}}{\partial r}}{\frac{\partial^{2}B_{r}}{\partial r^{2}}}}} & (23) \\{r \approx {1.5\frac{\frac{\partial B_{r}}{\partial r}}{\frac{\partial^{2}B_{r}}{\partial z^{2}}}}} & (24)\end{matrix}$

The performance of these functions (equations 17 through 24) may beestimated using the model of the magnetized target shown on FIG. 4. Atransform may be developed to convert the ratio to its correspondingactual distance. The ratios between a magnetic field intensity and afirst spatial derivative of the magnetic field (given in equations 17and 18) are evaluated in the plots shown on FIGS. 8A and 8B. FIG. 8Adepicts a plot of the ratio

$\frac{B_{r}}{\frac{\partial B_{z}}{\partial z}}$

versus actual distance at axial positions of 0.5, 1.5, and 2.5. In thisexample the ratio seems to be poorly suited to determining distance asit is substantially independent of distance. FIG. 8B depicts a plot ofthe ratio

${- 2}\frac{B_{r}}{\frac{\partial B_{r}}{\partial r}}$

versus actual distance at axial positions of 0.5, 1.5, and 2.5. In thisexample, the ratio varies monotonically with distance. The separationbetween the two curves at larger distances indicates that the ratio maybe somewhat sensitive to the absolute intensity of the magnetic poles.

The ratios between a magnetic field intensity and a second spatialderivative of the magnetic field (given in equations 19 and 20) areevaluated at normalized axial positions of 0.5, 1.5, and 2.5 in theplots shown on FIGS. 9A and 9B. FIG. 9A depicts a plot of the ratio

$\left( {6\frac{B_{r}}{\frac{\partial^{2}B_{r}}{\partial r^{2}}}} \right)^{\frac{1}{2}}$

versus actual distance while FIG. 9B depicts a plot of the ratio

$\left( {{- 3}\frac{B_{r}}{\frac{\partial^{2}B_{r}}{\partial z^{2}}}} \right)^{\frac{1}{2}}$

versus actual distance. In these examples, the ratios vary monotonicallywith distance and may therefore be suitable for use in distancedetermination. The separation between the two curves in each plotindicates that these ratios may be somewhat sensitive to the absoluteintensity of the magnetic poles.

The ratios between a first spatial derivative of the magnetic field anda second spatial derivative of the magnetic field (given in equations 21through 24) are evaluated at normalized axial positions of 0.5, 1.5, and2.5 in the plots shown on FIGS. 10A, 10B, 10C, and 10D. FIG. 10A depictsa plot of the ratio

$6\frac{\frac{\partial B_{z}}{\partial z}}{\frac{\partial^{2}B_{r}}{\partial r^{2}}}$

versus actual distance. In this example the ratio is a strong monotonicfunction of the distance making it a good candidate for distancedetermination. FIG. 10B depicts a plot of the ratio

${- 3}\frac{\frac{\partial B_{z}}{\partial z}}{\frac{\partial^{2}B_{r}}{\partial z^{2}}}$

versus actual distance. The second spatial derivative in this ratio mayalso be determined by measuring ∂/∂z (∂B_(z)/∂r) or ∂/∂r(∂B_(z)/∂z).FIG. 10C depicts a plot of the ratio

${- 3}\frac{\frac{\partial B_{r}}{\partial r}}{\frac{\partial^{2}B_{r}}{\partial r^{2}}}$

versus actual distance. In these examples, the ratios vary monotonicallywith distance and may therefore be suitable for use in distancedetermination. The separation between the two curves in FIGS. 10A and10B indicates that these ratios may be somewhat sensitive to theabsolute intensity of the magnetic poles. The ratio in FIG. 10C showsvery little sensitivity to the absolute intensity of the magnetic poles.FIG. 10D depicts a plot of the ratio

$1.5\frac{\frac{\partial B_{r}}{\partial r}}{\frac{\partial^{2}B_{r}}{\partial z^{2}}}$

versus actual distance. The second spatial derivative in this ratio mayalso be determined by measuring ∂/∂r (∂B_(z)/∂z) or ∂/∂z (∂B_(z)/∂r). Inthis example, the ratio is not well correlated with distance.

It will be understood that method 100 may be performed using upholeand/or downhole processors. The disclosed embodiments are not limited inthis regard. For example, magnetic field measurements may be transmittedto the surface (using any suitable telemetry techniques). The distancemay then be computed at the surface and further used to compute a newdrilling direction which may then be transmitted back to the tool.Alternatively, the magnetic field measurements may be processed downholeto obtain the distance, for example, using one or more look up tables tocorrelate the computed ratio(s) to distance. The obtained distance maythen be used to compute a new drilling direction downhole which may beimplemented as part of a closed loop well twinning methodology.

While the aforementioned examples make use of a target well ismagnetization having axially spaced opposing magnetic poles it will beunderstood that the disclosed embodiments are not so limited. The use offirst spatial derivatives and second spatial derivatives of the magneticfield and ratios including those derivatives may be used withsubstantially any suitable target well magnetization.

Although a method for magnetic gradient and curvature based ranging andcertain advantages thereof have been described in detail, it should beunderstood that various changes, substitutions and alternations can bemade herein without departing from the spirit and scope of thedisclosure as defined by the appended claims.

What is claimed is:
 1. A method for determining a distance from adrilling well to a magnetized target well, the method comprising: (a)deploying a drill string in the drilling well, the drill stringincluding at least one magnetic field sensor in sensory range ofmagnetic flux emanating from the magnetized target well; (b) making aplurality of spaced apart magnetic field measurements in the drillingwell; (c) processing the spaced apart magnetic field measurements toobtain a ratio of a magnetic field intensity to a first spatialderivative of a magnetic field; and (d) processing the ratio computed in(c) to obtain the distance from the drilling well to the magnetizedtarget well.
 2. The method of claim 1, wherein the target well ismagnetized such that it includes a substantially periodic pattern ofopposing north-north (NN) magnetic poles and opposing south-south (SS)magnetic poles spaced apart along a longitudinal axis thereof.
 3. Themethod of claim 2, wherein the plurality of spaced apart magnetic fieldmeasurements are made in (b) at locations adjacent to one of theopposing NN or SS magnetic poles.
 4. The method of claim 1, wherein: themagnetic field measurements made in (b) are radially spaced apart; andthe magnetic field measurements are processed in (c) to obtain a ratioof the magnetic field intensity of a radial component of the magneticfield to the first spatial derivative of the radial component of themagnetic field in the radial direction.
 5. The method of claim 1,wherein: the magnetic field measurements made in (b) are axially spacedapart; and the magnetic field measurements are processed in (c) toobtain a ratio of the magnetic field intensity of a radial component themagnetic field to the first spatial derivative of an axial component ofthe magnetic field in the axial direction.
 6. The method of claim 1,further comprising: (e) processing the magnetic field measurements madein (b) to compute a tool face to target direction.
 7. A method fordetermining a distance from a drilling well to a magnetized target well,the method comprising: (a) deploying a drill string in the drillingwell, the drill string including a magnetic field sensor in sensoryrange of magnetic flux emanating from the magnetized target well; (b)making a plurality of spaced apart magnetic field measurements in thedrilling well; (c) processing the spaced apart magnetic fieldmeasurements to obtain a ratio of a magnetic field intensity to a secondspatial derivative of a magnetic field; and (d) processing the ratiocomputed in (c) to obtain the distance from the drilling well to themagnetized target well.
 8. The method of claim 7, wherein the targetwell is magnetized such that it includes a substantially periodicpattern of opposing north-north (NN) magnetic poles and opposingsouth-south (SS) magnetic poles spaced apart along a longitudinal axisthereof.
 9. The method of claim 8, wherein the plurality of spaced apartmagnetic field measurements are made in (b) at locations adjacent to oneof the opposing NN or SS magnetic poles.
 10. The method of claim 17,wherein: the magnetic field measurements made in (b) are radially spacedapart; and the magnetic field measurements are processed in (c) toobtain a ratio of the magnetic field intensity of a radial component themagnetic field to the second spatial derivative of the radial componentof the magnetic field in the radial direction.
 11. The method of claim10, wherein: the magnetic field measurements made in (b) are axialspaced apart; the magnetic field measurements are processed in (c) toobtain a ratio of the magnetic field intensity of a radial component themagnetic field to the second spatial derivative of the radial componentof the magnetic field in the axial direction.
 12. The method of claim10, further comprising: (e) processing the magnetic field measurementsmade in (b) to compute a tool face to target direction.
 13. A method fordetermining a distance from a drilling well to a magnetized target well,the method comprising: (a) deploying a drill string in the drillingwell, the drill string including a magnetic field sensor in sensoryrange of magnetic flux emanating from the magnetized target well; (b)making a plurality of spaced apart magnetic field measurements in thedrilling well; (c) processing the spaced apart magnetic fieldmeasurements to obtain a ratio of a first spatial derivative of themagnetic field and a second spatial derivative of the magnetic field;and (d) processing the ratio computed in (c) to obtain the distance fromthe drilling well to the magnetized target well.
 14. The method of claim13, wherein the target well is magnetized such that it includes asubstantially periodic pattern of opposing north-north (NN) magneticpoles and opposing south-south (SS) magnetic poles spaced apart along alongitudinal axis thereof.
 15. The method of claim 14, wherein theplurality of spaced apart magnetic field measurements are made in (b) atlocations adjacent to one of the opposing NN or SS magnetic poles. 16.The method of claim 13, wherein: the magnetic field measurements made in(b) are radially spaced; and the magnetic field measurements areprocessed in (c) to obtain a ratio of the first spatial derivative of aradial component of the magnetic field in the radial direction to thesecond spatial derivative of the radial component of the magnetic fieldin the radial direction.
 17. The method of claim 13, wherein: themagnetic field measurements made in (b) are axially spaced; and themagnetic field measurements are processed in (c) to obtain a ratio ofthe first spatial derivative of an axial component of the magnetic fieldin the axial direction to the second spatial derivative of a radialcomponent of the magnetic field in the axial direction.
 18. The methodof claim 13, wherein: the magnetic field measurements made in (b) areboth axially spaced and radially spaced; the magnetic field measurementsare processed in (c) to obtain a ratio of the first spatial derivativeof an axial component of the magnetic field in the axial direction tothe second spatial derivative of a radial component of the magneticfield in the radial direction.
 19. The method of claim 13, wherein: themagnetic field measurements made in (b) are both axially spaced andradially spaced; the magnetic field measurements are processed in (c) toobtain a ratio of the first spatial derivative of a radial component ofthe magnetic field in the radial direction to the second spatialderivative of the radial component of the magnetic field in the axialdirection.
 20. The method of claim 13, further comprising: (e)processing the magnetic field measurements made in (b) to compute a toolface to target direction.